**Date & Time:** October 28, 2015, 15:00-16:00.

**Venue:** Ramanujan Hall

**Title:** Invariants of several matrices under SL(n) \times SL(n)-action.

**Speaker:** K. V. Subrahmanyam, CMI Chennai

**Abstract:** Let $R(m,n)$ denote the ring of invariant polynomial functions of the $
SL(n) \times SL(n) action $ on $m$ tuples of matrices. We describe the
ring of relations (the second fundamental theorem) among these
invariants. We also describe the $S(m) \times S(m)$-module structure of the
invariant ring, where $S(m)$ denotes the symmetric group on [m], and also the
module structure for the diagonal action of $S(m)$. We describe another natural
relation among the invariants which we believe will be useful to give an upper
bound on the degree in which the ring of invariants is generated.
We give an algorithm to detect when a tuple of matrices is in the null
cone of this action, which runs in time polynomial in the degree in
which the invariant ring is generated. This algorithm rides on an
algorithm of Gurvits, and our analysis is based on a novel use of
blow-ups of matrices, which we will outline. We will also state some
recent developments, a polynomial time algorithm for the same problem by two
independent groups of researchers.
This is joint work with Gabor Ivanyos and Youming Qiao.